The Dirichlet problem for perturbed Stark operators in the half-line

2022/05/19 by Toloza, Julio H., Uribe, Alfredo · 1 citation
#34L15 (Primary) 34E10 #81Q05 #81Q10 (Secondary) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2205.09275

Abstract

We consider the perturbed Stark operator H_qφ= -φ" + xφ+ q(x)φ, φ(0)=0, in L²(ℝ₊), where q is a real-valued function that belongs to \mathfrakA_r =\ q\inA_r\capAC[0,∞) : q'\inA_r\, where A_r = L²(ℝ₊,(1+x)^r dx) and r>1 is arbitrary but fixed. Let \λ_n(q)\_n=1^ ∞ and \κ_n(q)\_n=1^ ∞ be the spectrum and associated set of norming constants of H_q. Let \a_n\_n=1^∞ be the zeros of the Airy function of the first kind, and let ω_r:ℕ→ℝ be defined by the rule ω_r(n) = n^-1/3log^1/2n if r∈(1,2) and ω_r(n) = n^-1/3 if r∈[2,∞). We prove that λ_n(q) = -a_n + π(-a_n)^-1/2∫₀^∞ Ai²(x+a_n)q(x)dx + O(n^-1/3ω_r²(n)) and κ_n(q) = - 2π(-a_n)^-1/2∫₀^∞ Ai(x+a_n)Ai'(x+a_n)q(x)dx + O(ω_r³(n)), uniformly on bounded subsets of \mathfrakA_r. In order to obtain these asymptotic formulas, we first show that λ_n:A_r→ℝ and κ_n:A_r→ℝ are real analytic maps.

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• Asymptotics of the spectral data of perturbed Stark operators in the half-line with mixed boundary conditions

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